Slow relaxation and compaction of granular systems

0.65
0.64
0.63
0.62
0.61
0.6
0.59
0 1 2 3 4 5 6 7
Γ
More recently, Philippe and Bideau 11 carried out
compaction experiments; in the following we will
refer to these as the Rennes group’s experiments.
The vessel used is a 10-cm-diameter cylinder and
1-mm-diameter glass beads, which leads to 100
grains between the side walls. This restricts the
boundary effects but, contrary to the Chicago group’s
experiments, allows convection. The packing fraction
is measured using a γ-ray absorption set-up 11 . The
relaxation laws obtained by these authors differ
signifi cantly from those obtained by Knight et al. 9 ,
especially for the long-time behaviour. Indeed,
whereas in previous experiments no clear steadystate
was reached, this is defi nitely established in
Rennes group’s experiments, and may correspond
to a dynamical balance between convection and
compaction. The relaxation is better fi tted by the
Kohlrausch–Williams–Watts law (KWW law) — a
stretched exponential:
ρ(t) = ρ f – (ρ f – ρ 0) exp[–(t/τ) β ] (2)
where ρ f and ρ 0 correspond respectively to the
steady-state and to the initial packing-fraction value.
The adjustable parameters τ and β correspond here
respectively to the relaxation time and to the stretching
of the exponential. This characteristic timescale is found
to be well described by an Arrhenius behaviour τ = τ 0
exp [Γ 0/Γ]. Such a relaxation law is also found for strong
glasses (the dimensionless acceleration Γ plays the role
of the temperature).
Another quantity of interest is the fi nal packing
fraction obtained by the fi t: ρ f. Here again there
exists a strong discrepancy between the Chicago
group’s experiments and the Rennes group’s work.
In the former case this packing fraction is found to
increase with the tapping intensity, Γ, whereas it is
found to decrease in the latter. These discrepancies
are related to the fact that the Chicago experiments
are performed in a region where the system is far
from stationary, whereas the Rennes experiments are
focusing on the stationary regime. This is likely to
originate from the difference of confi nement between
the two experiments. Indeed, for the Chicago group’s
work, the strong boundary effects lead to order
ρ
creation at least close to the side walls and to packing
fractions higher than 0.64. On the contrary, all the
fi nal packing fractions obtained for glass beads and
with a very low confi nement by the Rennes group
are below this value. Nevertheless, these results are
affected by convection. Indeed, a signifi cant change
is observed in the dependence of ρ f on Γ, which
might correspond to different convective regimes 11,12 .
Under a threshold Γ c ≈ 2, the fi nal state of the free
surface of the packing is an inclined
plane and indicates a spontaneous breaking of
symmetry. Above this value, the free surface takes a
fl at conical shape.
It should be pointed out that all the studies
reviewed above deal with isotropic granular
media. Of course, most actual granular materials
are far from being isotropic, and the grain shape
may modify the behaviour of the system during
compaction. Villarruel et al. 13 have shown, using
a Chicago set-up, that a nematic ordering can be
observed for compaction of rods. Ribière et al.
(ref. 14, and ibid, manuscript in preparation) carried
out experiments of compaction of rice with the
set-up in Rennes (low confi nement). They did not
observe such an ordering and obtained compaction
characteristics similar to those obtained with glass
beads. Note that the aspect ratio of the grains is
probably an important parameter of the problem.
ANNEALING AND MEMORY EFFECTS
Further insight into the understanding of the nature
of the relaxation can be gained by allowing the tap
intensity to vary in time. Nowak et al. 10,15 (and later
Philippe 16 with the Rennes set-up) have reported
annealing experiments. Using the Chicago set-up they
proceeded as follows: starting from a loose packing of
grains, the material is tapped at a given intensity Γ for
a given time t (10 5 taps). Γ is then modifi ed and the
compaction process continued for t taps (see Fig. 2a).
The increase of Γ corresponds to an increase of the
average packing fraction except for values larger than
three for which a slow decrease can be observed. This
slow decrease can be interpreted as void creation
due to a too-large agitation. If Γ is then reduced, the
PROGRESS ARTICLE
0.66
a b Figure 2 Annealing and
ρ
Reversible
Irreversible
Increasing
Decreasing
Increasing again
0.62
0.615
0.61
–10 0 10 20 30 40 50
t–t 0
aging during compaction.
a, Annealing curve. The initial
packing was prepared in a
low-density initial confi guration
(ρ ≈ 0.59) and then the
acceleration amplitude Γ was
slowly fi rst increased (solid
circles) and then decreased
(open circles). At each value
of Γ the system was tapped
10 5 times and Γ incremented
by ΔΓ ≈ 0.5. The higher
density branch (the upper
one) is reversible to changes
in Γ (see square symbols).
Each curve is an average of
separate experimental runs
and the error bars represent
the r.m.s. variations between
runs. Reprinted from ref. 10,
Copyright (1997), with
permission from Elsevier.
b, Time evolution of packing
fraction for a system that was
compacted to ρ 0 = 0.613 at
time t 0 using three different
accelerations: Γ 1 = 1.8 (circles),
Γ 0 = 4.2 (triangles) and
Γ 2 = 6.3 (diamonds).
After the packing fraction
ρ 0 was achieved (t = 0),
the system was vibrated at
acceleration Γ 0. The evolution
for t > t 0 depended strongly
on the previous history.
Reprinted from ref. 18.
Copyright (2000) by the
American Physical Society.
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